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By now you are comfortable with the graphical implications of the first and second derivatives of a function. gives you the slope of the tangent while tells you the concavity of the graph at a point. There is nothing to stop us from finding the third and fourth (and beyond!) derivatives, but there are not any significant graphical characteristics associated with higher derivatives. is simply the rate of change of and so on.

Notation: We cannot simply put more and more tic marks for, say, the 7th derivative. Instead, the 7th derivative of is written , using italicized roman numerals. The th derivative of a function is written .

  1. Find for .

    Take the derivative four times.

  2. If , find an expression for the th derivative, .

    Take a few derivatives of this function and look for patterns.